### Student Page1.7.2Rectangle Patterns in the 100s Chart A4US

In Analyzing the 100s Chart you found patterns in rows, columns, or diagonals. This time your pattern search will be confined to rectangles.

#### 1.

Consider the rectangle in TableĀ 1.7.2.1. What patterns do you see in the numbers in this rectangle?

Let $$x = 84\text{.}$$ In the rectangle, write the other values in the rectangle in terms of $$x\text{.}$$

##### (a)

Use your relabeling to justify any numeric patterns you found.

##### (b)

Find another pattern in the relabeled rectangle.

##### (c)

Does your pattern work for a 4 by 3 rectangle with upper left corner labeled 51?

##### (d)

Does your pattern work for any other 4 by 3 rectangles? Why or why not?

Work on the the following exercises alone for at least 10 minutes. When each member of your group has found at least one rectangle pattern, share the patterns you found with your group.

#### 2.

Using a different color to outline each rectangle, draw three rectangles following the lines in TableĀ 1.6.3.1 (reprinted below) with the following rules:

 100 101 102 103 104 105 106 107 108 109 90 91 92 93 94 95 96 97 98 99 80 81 82 83 84 85 86 87 88 89 70 71 72 73 74 75 76 77 78 79 60 61 62 63 64 65 66 67 68 69 50 51 52 53 54 55 56 57 58 59 40 41 42 43 44 45 46 47 48 49 30 31 32 33 34 35 36 37 38 39 20 21 22 23 24 25 26 27 28 29 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 9
##### (a)

Make sure all three rectangles have different dimensions.

##### (b)

Make one rectangle large and one rectangle small.

##### (c)

Make only one square.

#### 3.

What patterns do you notice among the numbers in one of the rectangles? Find at least three patterns. Each pattern you find must not extend beyond the rectangle.

#### 4.

Of the patterns you found, do any work for all three rectangles? Explain.

#### 5.

Choose one of the patterns you found. Explain why the pattern works.

#### 6.

Challenge yourself to use variables to show why the pattern works.

As a group, choose your 3 favorite patterns, including at least one that you think other groups won't find. Share your patterns with the class. As a group, choose 2 patterns shared by other groups to justify. Use variables and algebra to justify that each pattern works.